On the two symmetries in the theory of m-Hessian operators

TL;DR

This paper explores the symmetries in m-Hessian operators, revealing new inequalities and variational problem solutions that deepen understanding of fully nonlinear operators.

Contribution

It introduces a novel perspective on the symmetries of m-Hessian operators, leading to new inequalities and simplified proofs of classical results.

Findings

An isoperimetric variational problem for Hessian integrals is formulated.

A new proof of Poincare-type inequalities for Hessian integrals is provided.

A new set of inequalities generated by specific functions is discovered.

Abstract

We show that the modern theory of fully nonlinear operators had been started by the skew symmetry of minors in cooperation with the symmetry of symmetric functions. The paper presents some consequences of this interaction for the m-Hessian operators. One of them is a setting of the isoperimetric variational problem for Hessian integral. The m-admissible minimizer is found, what brings out a new simple proof of the well known Poincare - type inequalities for Hessian integrals. Also a new set of inequalities, generated by a special finite set of functions, is found.